We propose the application of two well-established optimization methods not widely used in molecular mechanics to the solution of problems in protein structure prediction. The first problem, which has been addressed by other authors, as well as by the P.I. in previous work, is that of finding the native structure of a loop which leaves and re-enters a core region with known base geometry. Most other approaches have been variants of an "explore then minimize" strategy. We propose instead to use the technique of constrained minimization, in which one seeks to minimize a function subject to constraints among the independent variables. In this case, energy will be minimized subject to the condition that the base geometry be correct; thus it involves a one-step closure with minimization. This has potential to be faster than other strategies. We will use Fortran routines from the Harwell Subroutine Library. After experimentation with the convergence behavior of two routines (VF01A and VF13AD) on a test surface (F=(sin x)(sin y)), we will choose one of them to apply to the determination of the structure of the hypervariable loops in an immunoglobulin, as a test case. The speed and efficiency of the method will be compared to earlier results obtained by the P.I. for the same system using a "search then minimize" approach. Another problem in homologous modeling of proteins is predicting the side- chain conformations in a region of the protein where the backbone is known. The side-chains in the interiors of globular proteins exist in generic conformations, or "rotamers," and several authors have assembled "rotamer libraries." We ask: How can we predict the native side-chain configuration (list of rotamers), given a sequence, a rotamer library, and a backbone tertiary structure. In previous work we devised a simple a-priori scoring criterion for rotamer configurations (based on number of interatomic contacts in the configuration and frequency of appearance of the rotamers) and we showed by exhaustive search that this criterion ranks the native structure very high up in the list of configurations. Now we wish to first, apply methods of combinatorial optimization -- specifically, simulated annealing -- in the discrete rotamer space, in order to seek out promising configurations for large cores, and second, to search for criteria which will distinguish the native configuration from its high- ranking brethren. The method will be tested and parameterized using the cores already studied (up to about 107 configurations); we will then apply it to larger cores. Simultaneously, we will look at sets of high-ranking configurations using energy minimization, buried hydrophobic surface measurements and determinations of packing density in order to determine whether any of these criteria help in identifying the native configuration.